1. Technical Field
The present invention relates to forecasting and inventory control, and more particularly, relates to forecasting intermittent demand.
2. Related Art
In many of today's competitive commercial activities, such as manufacturing and retail, inventory management is increasingly becoming recognized as one of the most important operational concerns. In particular, by maintaining precise inventory levels, a business entity can eliminate the need to manufacture, purchase, and/or store goods that are not immediately required. In order to achieve such goals, an inventory management system must have the ability to accurately forecast demand in order to predict inventory requirements.
One particular type of demand forecasting that is especially difficult to predict involves that of "intermittent demand," which is characterized by frequent zero values intermixed with nonzero values. Intermittent demand patterns are found in various situations, such as those dealing with spare parts, or the sale of "big ticket" items. A typical intermittent demand data set will generally represent a series of requirements for a particular part over a period of time. For example, a parts supplier will typically track a monthly demand for each of their parts. Over a particular time span, for example 18-36 months, it may be apparent that demand for each part exists only during a few of the months. Thus, the demand history, on a month-by-month basis, would reflect that most of the monthly demand values were zero, while a small minority were nonzero. Because most intermittent demand series have no trend or seasonality, and because the nonzero values can greatly vary, forecasting future demand based on such historical data has in the past involved nothing more than oversimplified and relatively inaccurate statistical methods. In turn, this makes it difficult to provide an effective inventory management system.
In general, demand forecasting involves using historical data to predict future requirements over some "lead time" L, for example, four months. The predicted requirements over the lead time is generally provided as a distribution and referred to as the lead time demand (LTD). Existing techniques used to forecast intermittent demand include single exponential smoothing and Croston's method. Single exponential smoothing forecasts assume the distribution of lead time demand to be described by a normal (i.e., Gaussian or "bell-shaped") curve and use a smoothing process to estimate the mean and variance of the distribution. The total demand is regarded as normal, being the sum of L random variables that are independent and identically distributed. Of course, the normality assumption is unlikely to be strictly true, especially for short lead times that do not permit central limit theorem effects for the sum.
Below is an example of one possible method for implementing single exponential smoothing. In this example, the mean and variance of the assumed normal distribution of LTD may be computed as follows. The mean level of demand at time t, M(t), may be computed using EQU M(t)=.alpha.X(t)+(1-.alpha.)M(t-1), t=1 . . . T,
where X(t) is the observed demand in period t and .alpha. is a smoothing constant between 0 and 1. For each inventory item, .alpha. may be computed, for example, by a grid search in steps (e.g., 0.05), and then selecting the value of .alpha. that minimized the sum of squared residuals .SIGMA.(X(t)-M(t)).sup.2, t=1 . . . T. The smoothing may then be initialized using the average of the first two demands, EQU M(0)-(X(1)+X(2))/2.
The mean of the L demands over the lead time is estimated as L.multidot.M(T). The common variance, V, is estimated from the one step ahead forecast errors using EQU V=(1/T).multidot..SIGMA.(X(t)-M(t-1)).sup.2, t=1 . . . T.
Finally, assuming independent and identically distributed demands, the variance of the LTD distribution is estimated as L.multidot.V.
Croston's method was developed to provide a more accurate estimate of the mean demand per period. Given this alternative estimate of the mean and the resulting estimate of the variance, Croston's method also assumes that LTD has a normal distribution.
Croston's method estimates the mean demand per period by applying exponential smoothing separately to the intervals between nonzero demands and their sizes. Let I(t) be the smoothed estimate of the mean interval between nonzero demand. Let S(t) be the smoothed estimate of the mean size of a nonzero demand at time t. Also let q be a counter for the interval between nonzero demands. Croston's method works as follows: EQU If X(t)=0, then EQU S(t)=S(t-1) EQU I(t)=I(t-1) EQU q=q+1,
else EQU S(t)=.alpha.X(t)+1-.alpha.)(S(t-1) EQU I(t)=.alpha.q+(1-.alpha.)I(t-1) EQU q=1,
where X(t) is the observed demand at time t. Combining the estimates of size and interval provides the estimate of mean demand per period EQU M(t)=S(t)/I(t).
These estimates are only updated when demand occurs. When demand occurs every review interval, Croston's method is identical to conventional exponential smoothing, with S(t)=M(t). Croston's method can be initialized using the time until the first event and the size of the first event. As with exponential smoothing, the mean of the L demands is estimated over the lead time as L.multidot.M(T), the variance of the LTD distribution is L.multidot.V, and LTD is assumed normal.
While these methods have provided some solution to the forecasting of intermittent data, the level of accuracy remains fairly low in most real world applications. In particular, these methods focus on estimating the mean demand, and generally provide only an inaccurate estimate of the entire forecast distribution. Accordingly an improved system for forecasting intermittent demand is required.